{"id":"openai-math/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs","pdf":"https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026","tags":["percolation","probability","graph-theory"],"score":0,"title":"Nonuniqueness of percolation on nonamenable quasi-transitive graphs","domain":"Probability and statistical mechanics","family":"214","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's stronger operator-threshold conjecture, establishing $`p_c\\lt p_{2\\to2}\\le p_u`$.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":3,"published":"2026-09-24","reactions":{},"discussion":[{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=for%20bond%20percolation%20on%20infinite%20connected%20locally%20finite%20quasi-transitive%20graphs","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"for bond percolation on infinite connected locally finite quasi-transitive graphs with , extending the setting of my recent post on an AI proof of critical nonpercolation . It also announces for nonamenable graphs in this class , resolving the Benjamini–Schramm nonuniqueness conjecture discussed…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"},{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=The%20converse%20is%20true%20by%20work%20of%20Burton%20and%20Keane","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"The converse is true by work of Burton and Keane. I loved this problem so deeply. It said something so beautiful and simple about these highly treelike graphs which had somehow eluded proof in the general case. It’s why I fell in love with percolation theory. Unsurprisingly the OpenAI preprint…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"},{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=whether%20there%20is%20a%20more%20natural%20proof%20for%20groups%20with%20free%20subgroups","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"whether there is a more natural proof for groups with free subgroups. At the same time, this problem deserved more endeavors. There were many ways ahead, and we could explore different routes to eventually reach the destination. It could be more humane. The journey could be enjoyed by people. But…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"}],"formalized":false,"family_title":"The Benjamini–Schramm nonuniqueness conjecture"}